How to write this as a boolean expression?

Question

How can I write the following sentence as boolean expression: $$ \text{If two sides of triangle are the same, then two opposite angles are the same} $$

I konw it should be something like this: $$ a = \text{if two sides of triangle are the same}\\ b = \text{two opposite angles are the same}\\ \\ a <\text{some logical operator}> b\\ $$

But I am not sure how to write $then$ as logical operator.

Thanks.

Answer 1

Let $\triangle ABC$ be a triangle with vertices at $A, B, $ and $C$. Denote the length of the side opposite angle $A$ by $a$, the length of the side opposite angle $B$ by $b$, and the length of the side opposite angle $C$ by $c$. Denote the measure of an angle $\theta$ by $m\angle \theta$.

Then $$(a = b) \rightarrow (m\angle A = m\angle B)$$ $$(a = c) \rightarrow (m\angle A = m\angle C)$$ $$(b = c) \rightarrow (m\angle B = m \angle C)$$

There are many variations for how you might express this relationship. I've specified "lengths" of sides and "measures" of angles to ensure we pin down what we mean by "the same:" Here, I'm assuming we are talking about sides that have equal length, and angles that have equal measures. The term that is most commonly used to pin down what we mean by "the same" in geometry is the term "congruent."

As you've formulated the propositions, we have simply "If $a$, then $b$: $\;a \rightarrow b$.

Answer 2

As for typography: Your $a\mathrel{\langle\text{some logical operator}\rangle}b$ can be written as either of $$ a\supset b \qquad a\to b \qquad a\Rightarrow b $$ There's no universal difference in meaning between these symbols. Educators at elementary levels (who teach logic as a tool rather than a separate object of study) almost always favor $\Rightarrow$, whereas proof theorists tend to prefer $\to$. The $\supset$ notation (originally an upside-down C, nothing to do with supersets) is found mostly in older logic texts, but not completely unknown in contemporary works either.